On the validation of fiducial techniques

TLDR: In this paper, the authors show that the techniques used by Fisher in deriving and handling his fiducial probability distribution are valid when dealing with a structured distribution, which is a family of random vectors defined on a probability space with values in a sample space.

Abstract: A structured model is essentially a family of random vectors Xθ defined on a probability space with values in a sample space. If, for a given sample value x and for each ω in the probability space, there is at most one parameter value θ for which Xθ(ω) is equal to x, then the model is called additive at x. When a certain conditional distribution exists, a frequency interpretation specific to additive structured models holds, and is summarized in a unique structured distribution for the parameter. Many of the techniques used by Fisher in deriving and handling his fiducial probability distribution are shown to be valid when dealing with a structured distribution. Un modele structure est essentiellement une famille de vecteurs Xθ definis dans un espace de probabilite et a valeurs dans un espace d'echantillons. Si, pour un echantillon x donne et pour chaque ω dans l'espace de probabilite, il existe au plus une valcur θ du parametre pour laquelle Xθω est egal a x, alors le modele est dit additif en x. Si, de plus, une certaine distribution conditionnelle existe, alors une interpretation en termes de frequence, laquelle est specifique aux modeles structures additifs, est possible et donne lieu a une unique distribution structuree. On montre aussi que plusieurs techniques utilisees par Fisher dans l'etablissement et la manipulation de ses probabilites fiduciarires sont valides lorsque appliquees a une distribution structuree.